Taylor Series for Functions of one Variable

where R_n, the remainder after n terms, is given by either of the following forms:
Lagrange’s form R_n = f⁽ⁿ⁾(x - a)ⁿCauchy’s form R_n =

The value ξ, which may be different in the two forms, lies between a and x. The result holds if f(x) has continuous derivatives of order n at least.

If lim_{n → ∞} R_n = 0, the infinite series obtained is called the Taylor series for f(x) about x = a. If a = 0 the series is often called a Maclaurin series. These series, often called power series, generally converge for all values of x in some interval called the interval of convergence and diverge for all x outside that interval.

Binomial Series

Secial cases are
(a + x)² = a² + 2ax + x²
(a + x)³ = a³ + 3a²x + 3ax² + x³
(a + x)⁴ = a⁴ + 4a³x + 6a²x² + 4ax³ + x⁴
(1 + x)^{- 1} = 1 - x + x² - x³ + x⁴ - ...        - 1 < x < 1
(1 + x)^{- 2} = 1 - 2x + 3x² - 4x³ + 5x⁴ - ....        - 1 < x < 1
(1 + x)^{- 3} = 1 - 3x + 6x² - 10x³ + 15x⁴ - ....        - 1 < x < 1

Series if Exponential and Logarithmic Functions

       - ∞ < x < ∞
       - ∞ < x <∞

Series of Trigonometric Functions

Series of Hyperbolic Functions

Miscellaneous Series

Reversion of Power Series

If
y = c₁x + c₂x² + c₃x³ + c₄x⁴ + c₅x⁵ + ....
then
x = C₁y + C₂y² + C₃y³ + C₄y⁴ + C₅y⁵ + ....
where
c₁C₂ = - c₂;
c₁⁵C₃ = 2c₂² - c₁c₃;
c₁⁷C₄ = 5c₁c₂c₃ - 5c₂³ - c₁²c₄;
c₁⁹C₅ = 6c₁²c₂c₄ + 3c₁²c₃² - c₁³c₃ + 14c₂⁴ - 21c₁c₂²c₁;
c₁¹¹C₆ = 7c₁³c₂c₅ + 84c₁c₂³c₃ + 7c₁³c₃c₄ - 28c₁²c₂c₃² - c₁⁴c₆ - 28c₁²c₂²c₄ - 42c₂⁵

Taylor Series for Functions of Two Variables

f(x, y) = f(a, b) + (x - a).f_x(a, b) + (y - b).f_y(a, b) + (1/2!).{(x - a)².f_xx(a, b) + 2(x - a)(y - b).f_xy(a, b) + (y - b)².f_yy(a, b)} + ....
where f_x(a, b), f_y(a, b) .... denote partial derivatives with respect to x, y, .... evaluated at x - a, y - b.